The truncated multidimensional moment problem: canonical solutions

Sergey M. Zagorodnyuk

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The truncated multidimensional moment problem: canonical solutions

Sergey M. Zagorodnyuk

Classical Analysis and ODEs · Submitted 28 Jun 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It is constructed a 1-1 correspondence between canonical solutions and flat extensions of the given moments (both sets may be empty).

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,304 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.37 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.26 · compares with prior results 0.05 · mentions code or data availability 0.04.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

For the truncated multidimensional moment problem we introduce a notion of a canonical solution. Namely, canonical solutions are those solutions which are generated by commuting self-adjoint extensions inside the associated Hilbert space. It is constructed a 1-1 correspondence between canonical solutions and flat extensions of the given moments (both sets may be empty). In the case of the two-dimensional moment problem (with triangular truncations) a search for canonical solutions leads to an algebraic system of equations. A notion of the index $i_s$ of nonself-adjointness for a set of prescribed moments is introduced. The case $i_s=0$ corresponds to flatness. In the case $i_s=1$ we get explicit necessary and sufficient conditions for the existence of canonical solutions. These conditions are valid for arbitrary sizes of truncations. In the case $i_s=2$ we get either explicit conditions for the existence of canonical solutions or a single quadratic equation with several unknowns. Numerical examples are provided.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	For the truncated multidimensional moment problem we introduce a notion of a canonical solution.	contribution 1.00	0.93	0.14	asserted 1.00
s1	Namely, canonical solutions are those solutions which are generated by commuting self-adjoint extensions inside the associated Hilbert space.	contribution 0.94	0.65	0.12	asserted 1.00
s2	It is constructed a 1-1 correspondence between canonical solutions and flat extensions of the given moments (both sets may be empty).	result 0.84	0.90	0.32	asserted 0.99
s3	In the case of the two-dimensional moment problem (with triangular truncations) a search for canonical solutions leads to an algebraic system of equations.	method 0.51	0.78	0.06	asserted 0.98
s4	A notion of the index $i_s$ of nonself-adjointness for a set of prescribed moments is introduced.	contribution 0.99	0.90	0.06	asserted 1.00
s5	The case $i_s=0$ corresponds to flatness.	interpretation 0.47	0.67	0.05	asserted 1.00
s6	In the case $i_s=1$ we get explicit necessary and sufficient conditions for the existence of canonical solutions.	result 0.93	0.91	0.18	asserted 1.00
s7	These conditions are valid for arbitrary sizes of truncations.	result 0.84	0.83	0.09	asserted 1.00
s8	In the case $i_s=2$ we get either explicit conditions for the existence of canonical solutions or a single quadratic equation with several unknowns.	result 0.94	0.93	0.11	asserted 0.79
s9	Numerical examples are provided.	contribution 0.45	0.92	0.03	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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The paper

Original abstract

For the truncated multidimensional moment problem we introduce a notion of a canonical solution. Namely, canonical solutions are those solutions which are generated by commuting self-adjoint extensions inside the associated Hilbert space. It is constructed a 1-1 correspondence between canonical solutions and flat extensions of the given moments (both sets may be empty). In the case of the two-dimensional moment problem (with triangular truncations) a search for canonical solutions leads to an algebraic system of equations. A notion of the index $i_s$ of nonself-adjointness for a set of prescribed moments is introduced. The case $i_s=0$ corresponds to flatness. In the case $i_s=1$ we get explicit necessary and sufficient conditions for the existence of canonical solutions. These conditions are valid for arbitrary sizes of truncations. In the case $i_s=2$ we get either explicit conditions for the existence of canonical solutions or a single quadratic equation with several unknowns. Numerical examples are provided.

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